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Mirrors > Home > MPE Home > Th. List > sbalex | Structured version Visualization version GIF version |
Description: Equivalence of two ways
to express proper substitution of a setvar for
another setvar disjoint from it in a formula. This proof of their
equivalence does not use df-sb 2075.
That both sides of the biconditional express proper substitution is proved by sb5 2276 and sb6 2095. The implication "to the left" is equs4v 2011 and does not require ax-10 2145 nor ax-12 2179. It also holds without disjoint variable condition if we allow more axioms (see equs4 2417). Theorem 6.2 of [Quine] p. 40. Theorem equs5 2461 replaces the disjoint variable condition with a distinctor antecedent. Theorem equs45f 2460 replaces the disjoint variable condition on 𝑥, 𝑡 with the nonfreeness hypothesis of 𝑡 in 𝜑. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2269 in place of equsex 2419 in order to remove dependency on ax-13 2373. (Revised by BJ, 20-Dec-2020.) Revise to remove dependency on df-sb 2075. (Revised by BJ, 21-Sep-2024.) |
Ref | Expression |
---|---|
sbalex | ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2156 | . . 3 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑡 → 𝜑) | |
2 | ax12v2 2181 | . . . 4 ⊢ (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))) | |
3 | 2 | imp 410 | . . 3 ⊢ ((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) |
4 | 1, 3 | exlimi 2219 | . 2 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) |
5 | equs4v 2011 | . 2 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) → ∃𝑥(𝑥 = 𝑡 ∧ 𝜑)) | |
6 | 4, 5 | impbii 212 | 1 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1540 ∃wex 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-10 2145 ax-12 2179 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ex 1787 df-nf 1791 |
This theorem is referenced by: equsexv 2269 sb5 2276 dfsb7 2284 mopick 2629 alexeqg 3550 dfdif3 4015 pm13.196a 41610 |
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